Geometry from a Differentiable ViewpointCambridge University Press, 22.10.2012 The development of geometry from Euclid to Euler to Lobachevsky, Bolyai, Gauss and Riemann is a story that is often broken into parts – axiomatic geometry, non-Euclidean geometry and differential geometry. This poses a problem for undergraduates: Which part is geometry? What is the big picture to which these parts belong? In this introduction to differential geometry, the parts are united with all of their interrelations, motivated by the history of the parallel postulate. Beginning with the ancient sources, the author first explores synthetic methods in Euclidean and non-Euclidean geometry and then introduces differential geometry in its classical formulation, leading to the modern formulation on manifolds such as space-time. The presentation is enlivened by historical diversions such as Huygens's clock and the mathematics of cartography. The intertwined approaches will help undergraduates understand the role of elementary ideas in the more general, differential setting. This thoroughly revised second edition includes numerous new exercises and a new solution key. New topics include Clairaut's relation for geodesics and the use of transformations such as the reflections of the Beltrami disk. |
Inhalt
Euclid | 12 |
Euclids theory of parallels | 19 |
The theory of parallels | 27 |
Similarity of triangles | 34 |
NonEuclidean geometry | 43 |
Neutral space | 55 |
Hyperbolic space | 62 |
The Elements Selections from Book XI | 72 |
Lengths angles and areas | 130 |
Map projections | 138 |
Geodesics | 185 |
The GaussBonnet Theorem | 201 |
Constantcurvature surfaces | 218 |
Abstract surfaces | 237 |
Modeling the nonEuclidean plane | 251 |
Where from here? | 282 |
Early Work on plane curves Huygens Leibniz Newton and Euler | 81 |
Involutes and evolutes | 89 |
Curves in space | 99 |
On Euclidean rigid motions | 110 |
Surfaces | 116 |
The tangent plane | 124 |
On the hypotheses which lie at | 313 |
Solutions to selected exercises | 325 |
Bibliography | 341 |
351 | |
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Häufige Begriffe und Wortgruppen
AABC abstract surface angle defect angle sum arc length asymptotic Axiom Beltrami chapter Christoffel symbols component functions compute congruent Consider construct coordinate chart coordinate curves cosh defined Definition denote derivative determined diffeomorphism differential equations differential geometry direction disk Euclid Euclidean expp expression figure find first fixed follows formula Gauss Gaussian curvature given line horocycle implies incidence geometry infinitely inner product interior angles intersection isometry Lemma line element line segment linear fractional transformation manifold map projection matrix metric relations non-Euclidean geometry open set parallel parametrization perpendicular polar coordinates Postulate proof properties Proposition prove quadrilateral reflection regular curve regular surface Riemann Riemannian metric right angles right triangle Saccheri satisfies Show side space sphere spherical stereographic projection straight lines subset Suppose surface in R3 surface of revolution tangent plane tangent vector tensor Theorem Tp(S U C R2 unit speed curve vector field